Distance between Excenters of Triangle in Terms of Circumradius/Proof
Jump to navigation
Jump to search
Theorem
Let $\triangle ABC$ be a triangle whose sides are $a$, $b$ and $c$ opposite vertices $A$, $B$ and $C$ respectively.
Let $I_b$ and $I_c$ be the excenters of $\triangle ABC$ with respect to $b$ and $c$ respectively.
Let $R$ be the circumradius of $\triangle ABC$.
Then:
- $I_b I_c = 4 R \cos \dfrac A 2$
Proof
From Triangle is Orthic Triangle of Triangle formed from Excenters, we establish that $\triangle ABC$ is the orthic triangle of $\triangle I_a I_b I_c$.
Hence $I_b B$ is an altitude of $\triangle I_a I_b I_c$.
Thus $\angle I_b B I_a$ is a right angle.
From Altitudes of Triangle Bisect Angles of Orthic Triangle:
- $\angle CBI = \dfrac B 2$
So:
- $\angle I_a B C = 90 \degrees - \dfrac B 2$
By a similar argument, mutatis mutandis:
- $\angle I_a C B = 90 \degrees - \dfrac C 2$
Hence:
\(\ds \angle B I_a C\) | \(=\) | \(\ds 180 \degrees - \paren {\angle I_a B C + \angle I_a C B}\) | Sum of Angles of Triangle equals Two Right Angles | |||||||||||
\(\ds \) | \(=\) | \(\ds 180 \degrees - \paren {\paren {90 \degrees - \dfrac B 2} + \paren {90 \degrees - \dfrac C 2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {B + C} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {180 \degrees - A} 2\) | Sum of Angles of Triangle equals Two Right Angles | |||||||||||
\(\ds \) | \(=\) | \(\ds 90 \degrees - \dfrac A 2\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds BC\) | \(=\) | \(\ds I_b I_c \map \cos {90 \degrees - \dfrac A 2}\) | Sides of Orthic Triangle of Acute Triangle | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds I_b I_c\) | \(=\) | \(\ds \dfrac a {\sin \frac A 2}\) | Cosine of Complement equals Sine | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \) | \(=\) | \(\ds \dfrac {2 R \sin A} {\sin \frac A 2}\) | Law of Sines | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \) | \(=\) | \(\ds \dfrac {2 R \cdot 2 \sin \frac A 2 \cos \frac A 2} {\sin \frac A 2}\) | Double Angle Formula for Sine | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \) | \(=\) | \(\ds 4 R \cos \frac A 2\) | Double Angle Formula for Sine |
$\blacksquare$
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: The ex-circles